Let \(\Omega\subset \mathbb C^n\), \(n>1\), be a bounded strictly pseudoconvex domain with smooth boundary \(\partial \Omega\). Fefferman proposed to study CR geometry of the boundary \(\partial \Omega\) via the formal asymptotics of the Dirichlet problem \begin{align*} &\mathcal J(u)=(-1)^n\operatorname{det}\left( \begin{array}{cc} u & u_{z^{\bar k}} \\ u_{z^j} & u_{z^jz^{\bar k}} \\ \end{array} \right)=1 \ \ \text {in} \ \ \Omega,\\ &u=0 \ \ \text{on} \ \ \partial \Omega,\\ &u>0 \ \ \text{in} \ \ \Omega. \end{align*} Cheng and Yau proved the existence of the unique solution \(u\) to Fefferman’s problem. Later Lee and Melrose showed that the solution \(u\) has an asymptotic expansion of the form \[ u\thicksim \rho\sum_{k=0}^{\infty}\eta_k\left(\rho^{n+1}\log \rho\right)^k, \ \eta_k\in \mathcal C^{\infty}(\bar \Omega), \] where \(\rho\) is a Fefferman defining function, e.g., a smooth defining function for \(\Omega\) satisfying \(\mathcal J(\rho)=1+O(\rho^{n+1})\). Graham showed that the coefficients \(\eta_k\) mod \(O(\rho^{n+1})\) are locally uniquely determined by \(\partial \Omega\) and are independent of the Fefferman defining function. The local invariant \(b\eta_1=\eta_1|_{\partial \Omega}\) of the boundary \(\partial \Omega\) is called the obstruction function.
A strictly pseudoconvex hypersurface \(M\) is called obstruction flat if the obstruction function for \(M\) vanishes. If \(M\) is locally CR equivalent to the unit sphere then we say that \(M\) is CR flat or locally (CR) spherical. There exist (local) real analytic strictly pseudoconvex hypersurfaces not locally spherical, for which the local invariant \(b\eta_1\) vanishes identically.
The goal of this paper is to consider the problem of determining whether the global vanishing of the obstruction function implies biholomorphic equivalence to the unit ball. The first main result is the following theorem. Let \(\Omega_t\), \(t\in [0,1]\), be a smooth family of smooth bounded strictly pseudoconvex domains in \(\mathbb C^2\), with \(\Omega_0\) the unit ball. If \(\partial \Omega_t\) is obstruction flat for all \(t\), then each \(\Omega_t\) is biholomorphic to the unit ball.
In the next result the authors prove that for more general deformations of the unit ball, the order of vanishing of the obstruction equals the order of vanishing of the CR curvature. Let \((S^3,H,J_0)\) be a CR 3-sphere and let \((S^3,H,J_t)\), \(t\in [0,\epsilon)\) be a smooth family of stably embeddable deformations. If the CR obstruction density \(\mathcal O_t\) of \((S^3,H,J_t)\) vanishes to order \(k\) at \(t=0\), so does the CR curvature tensor \(Q_t\).
Finally, the authors prove that for an abstract CR manifold with transverse symmetry, obstruction flatness implies local equivalence to the CR 3-sphere.